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In a quadratic function given in the form of \( ax^2 + bx + c \), the vertex of the parabola is one of the most crucial points. This point is where the parabola changes direction. For the exercise function \( f(x) = x^2 - 2x - 15 \), the vertex can be found using the vertex formula, \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -2 \). Substituting these values, we get \( x = 1 \). By substituting \( x = 1 \) back into the function, we find \( f(1) = -16 \). Hence, the vertex is located at \( (1, -16) \).

The vertex indicates the lowest point of the parabola since it opens upwards. If \( a \) were negative, the vertex would represent the highest point. Always remember that the vertex is a key element in sketching the graph of a quadratic function.

Graphing quadratic functions means plotting the curve that depicts the equation’s behaviour on a coordinate plane. Quadratics form a U-shaped curve known as a parabola. The symmetry of the parabola is crucial because it is always symmetric about a vertical line passing through its vertex.

For the function \( f(x) = x^2 - 2x - 15 \), the graph opens upward because the coefficient of \( x^2 \), which is \( 1 \), is positive. This means that as \( x \) moves away from the vertex in either direction, \( f(x) \) will increase, showing the upward-opening nature of the parabola. It is important to plot the vertex, x-intercepts, and y-intercept to sketch the graph accurately.

• Each point on the parabola corresponds to the set of \( (x, f(x)) \) solutions.
• The vertex \( (1, -16) \) serves as the axis of symmetry’s turning point.

Utilize these points to neatly draw and visualize the parabola.

X-intercepts are points where the graph of the function crosses the x-axis. At these points, the value of \( f(x) \) or \( y \) is zero. Finding these intercepts involves solving the equation \( f(x) = 0 \).

For \( f(x) = x^2 - 2x - 15 \), set the equation to zero to find x-intercepts: \( x^2 - 2x - 15 = 0 \). Factoring it, we get: \((x - 5)(x + 3) = 0\). Solving these factors gives us x-intercepts at \( x = 5 \) and \( x = -3 \). This means the graph crosses the x-axis at points \((5, 0)\) and \((-3, 0)\).

Understanding x-intercepts is essential, especially in real-world scenarios where they might represent potential solutions or outcomes. Graphically, they aid in defining the parabola's position relative to the axes.

A y-intercept is a point where the graph crosses the y-axis. At this juncture, the value of \( x \) is zero. Calculating the y-intercept of a quadratic function is straightforward: set \( x = 0 \) and compute \( f(0) \).

For \( f(x) = x^2 - 2x - 15 \), by setting \( x = 0 \), we find \( f(0) = 0^2 - 2 \times 0 - 15 = -15 \). Thus, the y-intercept is \((0, -15)\).

This point offers a valuable reference for drawing the parabola. On a graph, the y-intercept shows where the curve touches the y-axis, providing a clear starting point for further exploration of the quadratic function visually. Together with the vertex and x-intercepts, it enhances your understanding of the overall shape, direction, and position of the parabola.